Devices for generating nondeterministic, or true, random numbers are well known in the art. U.S. Pat. No. 6,763,364, issued Jul. 13, 2004, to Wilber, which is hereby incorporated by reference, describes one such true random number generator (TRNG) for use with a personal computer. U.S. Pat. No. 6,862,605, issued Mar. 1, 2005, to Wilber, which is hereby incorporated by reference, describes a TRNG comprising a low- and a high-frequency signal source. United States Patent Application Publication No. 2010/0281088, published Nov. 4, 2010, by Wilber, which is hereby incorporated by reference, describes a TRNG that is resettable in response to a trigger signal. United States Patent Application Publication No. 2016/0062735, published Mar. 3, 2016, by Wilber, which is hereby incorporated by reference, describes a quantum random number generator with quantifiable entropy.
Nondeterministic (i.e., true) random number generators are distinguished from pseudorandom number generators in that the future output numbers of the former are considered to be unpredictable in a real, theoretical sense, while the future output numbers of the latter are produced by algorithms that are completely predictable given knowledge of the algorithm design and its current state.
In addition to the broad categories of pseudorandom and nondeterministic random generators, generators that use quantum mechanical measurements to provide the entropy or nondeterminism are distinguishable from generators that measure highly complex or chaotic processes and only appear to be entirely unpredictable by statistical testing. The outcome of measurements of certain simple quantum mechanical systems can be shown mathematically to be non-computable, that is, they may have infinitely many possible states. In the context of quantum entropy sources, this concept is more formally treated by the Born rule, named after physicist Max Born. In a practical sense this means a sequence of numbers produced by such measurements is not only unpredictable, but subsequences will not repeat beyond statistical expectation regardless of how many numbers are produced. This is another way of saying its period, the quantity of numbers output before it begins to repeat, is unlimited or undefined.
Every pseudorandom generator has a finite and definite period, and the ultimate length of the period is limited by the complexity of the computer or device in which its program is running. That is because every computer is a finite state machine, which means, by definition, it can only take on a finite number of states before it must begin to repeat a previously produced pattern. Theoretically, every physical device can be considered a finite state machine because it is composed of a finite number of particles that can only take on a finite number of permutations or states. However, in a practical sense no physical device is a closed system, meaning that it may change over time in a fundamental way. This can occur in a number of ways, such as by the addition or loss of either energy, or mass in the form of the particles that compose it. In addition, for example in systems that measure thermal noise as their source of entropy, there are enough quantum mechanical interactions at the atomic level to keep such a device from behaving strictly as a classical device long enough ever to see a statistically significant pattern of repeating bits. A more thorough modeling of a thermal noise source, such as a resistor, includes a certain amount of parasitic capacitance. The parasitic capacitance appears in series with the resistor and forms, in the simplest model, a first order low-pass filter. Therefore, the voltage noise measured across the resistor's terminals has a finite bandwidth and a computable autocorrelation function (ACF). A sequence of random numbers produced from measurements of thermal noise has the same ACF as the thermal noise source. However, the transfer function—and resulting ACF—of the measurement circuitry usually limits how fast thermal noise sources can be sampled without introducing measurable autocorrelation. At a more fundamental level, some publications assert that thermal noise can be modeled as a chaotic system, albeit of high complexity, due to the very large number of charge carriers. It is further suggested, as would be the case with other classical chaotic systems, that with sufficient knowledge of the current states of the electrons in the resistor, and with enough computing power, the future value of the noise voltage could be predicted. Heisenberg's uncertainty principle precludes the possibility of accurately measuring both the positions and momenta of the electrons necessary to make such a prediction, so it is would appear theoretically impossible to accurately predict thermal noise of resistors of macroscopic scale beyond the level of determinism produced by the ACF noted above.
Random numbers used by most modern devices or applications, especially by computers or any systems containing microprocessors or other binary processing circuitry, are presented in the form of binary bits or binary encoded numbers. The sources of these random numbers most often produce them at specific intervals, resulting in what is generally called a time series. The statistical properties of a time series of random numbers be presented as a mean, a standard deviation (SD) and an autocorrelation function. The mean is typically directly related to the bias of a collection of bits, presented either as the probability of a “1” occurring, p(1), where 0.0≤p(1)≤1.0, or a fractional bias, BF=2p(1)−1, where −1.0≤BF≤1.0. The SD is a scaling factor that is not usually relevant to the quality of a random sequence. Finally, the ACF is a fundamentally important property of a random number sequence that quantifies correlations between bits in a sequence and other bits separated by various sampling intervals or orders. While only the bias and the ACF are necessary to specify the statistical properties of a sequence of binary random numbers, a number of statistical measures have been developed to look for specific patterns in such sequences. These specialized tests may reveal certain patterns more quickly and more dramatically.
The presence of patterns, along with their type and size, is one definition of statistical defects in random sequences, while the absence of patterns, meaning a fractional bias of 0.0 and an ACF of 0.0 at all orders, indicates a “perfect” random sequence. Of course, such a perfect random sequence only exists theoretically since it would have to contain an infinite number of bits to potentially satisfy these two requirements. In a practical sense, statistical defects in a random sequence produced by a real, physical generator can only be tested during a limited test period. It would be unrealistic to test a generator for many years since the generator's developer must either use it or make it available for sale on a reasonable timeline. Instead, statistical properties of a particular generator's output must be specified as limits, for example, |BF|≤10−8, or |ACF|≤10−8 for all orders up to 10,000. Asserted limits must be based both on large numbers of electronic and statistical tests and on a thorough understanding and mathematical modeling of the random number generation process. To illustrate why theoretical limits must be relied upon, the number of bits, n, required for direct statistical testing to a given confidence interval is n=p(1−p)(z/error)2, where the probability, p, is taken to be 0.5, z is the number of standard deviations that span the confidence interval in the normal distribution and error is the absolute deviation from the expected mean. For a 95% confidence interval, z=1.96 (for 99%, z=2.576). Given the example error of 10−8, the number of bits that must be tested to achieve a 95% confidence interval is 9.604×1015, or about 1/error2=1016 bits. Assuming a generation rate of 1 billion bits per second (1 Gbps), it would take over 3 years of continuous testing to complete. NIST defines “full entropy” for random bit generators in its Draft Special Publication 800-90B effectively as 1-ε bits/bit, where 0≤ε≤2−64. The specified lower limit on entropy is H=1-5.421011×1020, which is converted to a predictability by using a numerical inverse of the information entropy function, P=H−1. The calculated predictability is P=0.5+1.370686×10−10. The predictability of a perfectly random sequence is exactly 0.5. The error or difference from 0.5 is equivalent to an approximate upper limit of statistical defect, error=1.370686×10−10. Finally, the length of a random bit sequence needed to directly test for the level of statistical defect complying with NIST's full entropy definition with 95% confidence is 5.112×1019 bits. For an exemplary Gbps generator, the duration of testing could be 1,620 years.
Beyond the distinctions of quantum and chaotic sources of entropy, modern systems are being developed that are also concerned with other, more subtle effects of what is known as quantum nonlocality. Quantum nonlocality is a theory described by Albert Einstein and others that appears to show correlations of measurements in a physically separated system that cannot be simulated by classical mechanics or local hidden variable theories. This is what Einstein called, “spooky action at a distance.” While measurements of this effect are well documented by violations of Bell's inequality, they are still expected to be consistent with special relativity, meaning faster-than-light or superluminal communication of information is not expected.
The research, development and implementation of certain applications and products require the use of random numbers that can be shown to be theoretically and verifiably independent from other random numbers or quantum measurements. One example of such applications is the generation of free randomness or free random numbers. A free random number is a true random number that can only be correlated with events in its own future light cone. More specifically, a generator that outputs such a random number produces it during a known or definable generation period. From the beginning of the generation period, information associated with the random number, or the potential to produce correlations with it, can be thought of as moving outward from the generator at the speed of light. In special relativity, this is represented as a light cone, although from our perspective the theoretical field of influence seems to be moving outward from the generator equally in all three physical dimensions, creating a sphere. Any measurement made after the sphere has passed the physical location of said measurement can theoretically be influenced by or correlated with the random number. Measurements made outside this sphere cannot be influenced except by superluminal communication of information, which is believed to be disallowed by Einstein's relativity theory.
Other examples of technology or devices being developed that may require verifiably independent true random numbers are in the field of quantum cryptography, which may include a variety of tests for violation of Bell's inequality or other variations of tests incorporating demonstrations of quantum nonlocality. Quantum cryptography in its current form is primarily concerned with transmitting a cryptographic key between isolated locations while maintaining a high degree of confidence that the key has not been intercepted or copied by an attacker. This is known as quantum key distribution (QKD) and is commonly done by the use of entangled particles, especially photons, to transmit information. According to the principles of quantum mechanics, the particle being transmitted cannot be intercepted or read by an attacker without destroying the property of entanglement, thus alerting the intended receiver that the information has been compromised. More advanced or subtle variations of the simple implementations of quantum cryptography are known as device-independent quantum cryptography and post-quantum (or quantum resistant) cryptography. The former relies on tests demonstrating quantum nonlocality or so-called Bell tests, while the latter presupposes the existence of quantum computers and other advanced quantum-based eavesdropping technologies that put additional constraints on the design of cryptographic and quantum cryptographic systems.
Additionally, in the field of neuroscience, it is commonly believed that activity in the brain is solely responsible for the experience of consciousness or self-awareness and that no separate or external effects of mind are present. Whether or not such an assumption is true profoundly impacts some fundamental questions of our existence: the possibility or limitations of free will, mind-matter interaction and other anomalous effects and the possibility of some form of consciousness surviving physical death to name a few.